Pries: 676 Number Theory. 2010. Homework 8.
More about ramification
1. Let K = C(x) and L = K[y, z]/(y 2 − πx, z 3 − (y +
√
π)/(y −
√
π))
A. Show that Gal(L : K) = S3 .
B. Describe the primes of C[x] which are ramified in L.
2. If L|K is a Galois extension of number fields whose Galois group is non-cyclic, show
there are at most finitely many prime ideals for K which are non-split in L.
3. Suppose L|Q is a Galois extension. Suppose Q is a prime ideal of B = OL which
divides the prime ideal p of Z. Suppose p does not ramify in B. Show there is exactly
one automorphism σQ ∈ Gal(L : K) so that σ(b) ≡ bp mod Q for all b ∈ B. We call
σQ the Frobenius automorphism. Show that GQ is cyclic and generated by σQ .
4. Choose a topic for a 15 minute presentation. I’m happy to help brainstorm topics.